Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 143723, 6 pages
doi:10.1155/2008/143723
Research Article
The Periodic Character of the Difference Equation
xn1 fxn−l1, xn−2k1
Taixiang Sun1 and Hongjian Xi2
1
Department of Mathematics, College of Mathematics and Information Science, Guangxi University,
Nanning 530004, Guangxi, China
2
Department of Mathematics, Guangxi College of Finance and Economics, Nanning 530003,
Guangxi, China
Correspondence should be addressed to Taixiang Sun, stx1963@163.com
Received 3 February 2007; Revised 18 September 2007; Accepted 27 November 2007
Recommended by H. Bevan Thompson
In this paper, we consider the nonlinear difference equation xn1 fxn−l1 , xn−2k1 , n 0, 1, . . . ,
where k, l ∈ {1, 2, . . . } with 2k /
l and gcd 2k, l 1 and the initial values x−α , x−α 1, . . . , x0 ∈
0, ∞ with α max{l − 1, 2k − 1}. We give sufficient conditions under which every positive solution of this equation converges to a not necessarily prime 2-periodic solution, which extends and
includes corresponding results obtained in the recent literature.
Copyright q 2008 T. Sun and H. Xi. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In this paper, we consider a nonlinear difference equation and deal with the question of
whether every positive solution of this equation converges to a periodic solution. Recently,
there has been a lot of interest in studying the global attractivity, the boundedness character,
and the periodic nature of nonlinear difference equations e.g., see 1, 2. In 3, Grove et al.
considered the following difference equation:
xn1 p xn−2m1
,
1 xn−2r
n 0, 1, . . . ,
E1
where p ∈ 0, ∞ and the initial values x−α , x−α1 , . . . , x0 ∈ 0, ∞ with α max {2r, 2m 1},
and proved that every positive solution of E1 converges to not necessarily prime a 2speriodic solution with s gcd m 1, 2r 1. In 4, Stević investigated the periodic character
of positive solutions of the following difference equation:
xn1 1 xn−2s1
xn−2r1s1
,
n 0, 1, . . . ,
E2
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Advances in Difference Equations
and proved that every positive solution of E2 converges to not necessarily prime a 2speriodic solution, which generalized the main result of 5. Furthermore, Stević 6 studied the
periodic character of positive solutions of the following difference equation:
k
i1 αi xn−pi
xn 1 m
j1 βj xn−qj
,
n 1, 2, . . . ,
E3
where αi , i ∈ {1, . . . , k}, and βj , j ∈ {1, . . . , m}, are positive numbers such that Σki1 αi Σm
j1 βj 1, and pi , i ∈ {1, . . . , k}, and qj , j ∈ {1, . . . , m}, are natural numbers such that p1 < p2 < · · · < pk
and q1 < q2 < · · · < qm . For closely related results, see 7, 8.
In this paper, we consider the more general equation
xn1 f xn−l1 , xn−2k1 ,
n 0, 1, 2, . . . ,
1.1
where k, l ∈ {1, 2, . . . } with 2k /
l and gcd 2k, l 1, the initial values x−α , x−α1 , . . . , x0 ∈
0, ∞ with α max {l − 1, 2k − 1}, and f satisfies the following hypotheses:
H1 f ∈ CE×E, 0,∞ with a inf u,v∈E×E fu, v ∈ E, where E ∈ {0,∞, 0,∞};
H2 fu, v is decreasing in u and increasing in v;
H3 there exists a decreasing function g ∈ Ca, ∞, a, ∞ such that
i for any x > a, ggx x and x fgx, x;
ii lim x→a gx ∞ and lim x→∞ gx a.
The main result of this paper is the following theorem.
Theorem 1.1. Every positive solution of 1.1 converges to (not necessarily prime) a 2-periodic solution.
2. Proof of Theorem 1.1
In this section, we will prove Theorem 1.1. Without loss of generality, we may assume l < 2k
the proof for the case l > 2k is similar; then
{l, 2l, 3l, . . . , 2kl} {0, 1, 2, . . . , 2k − 1} mod 2k.
2.1
Lemma 2.1. Let {xn }∞
n−α be a positive solution of 1.1. Then there exists a real number L ∈ a, ∞
such that L ≤ xn ≤ gL for all n ≥ 1. Furthermore, let lim sup xn M and lim inf xn m, then
M gm and m gM.
Proof. By H1 and H2 , we have
xi f xi−l , xi−2k > f xi−l 1, xi−2k ≥ a
for every 1 ≤ i ≤ α 1.
2.2
Then there exists L ∈ a, ∞ with L < gL such that
L ≤ xi ≤ gL
for every 1 ≤ i ≤ α 1.
2.3
T. Sun and H. Xi
3
It follows from 2.3 and H3 that
gL f L, gL ≥ xα2 f xα2−l , xα2−2k ≥ f gL, L L.
2.4
Inductively, it follows that L ≤ xn ≤ gL for all n ≥ 1.
Let lim sup xn M and lim inf xn m, then there exist A, B, C, D ∈ m, M and sequences tn ≥ 1 and rn ≥ 1 such that
lim xtn M,
lim xtn −l A,
lim xtn −2k B,
n→∞
n→∞
n→∞
lim xrn m,
lim xrn −l C,
lim xrn −2k D.
n→∞
n→∞
2.5
n→∞
Thus by 1.1, H2 , and H3 , we have
f gM, M M fA, B ≤ fm, M,
f gm, m m fC, D ≥ fM, m,
2.6
from which it follows that gM ≥ m and gm ≤ M. Since g is decreasing, it follows that
m g gm ≥ gM,
M g gM ≤ gm.
2.7
Therefore, M gm and m gM. The proof is complete.
Proof of Theorem 1.1. Let {xn }∞
n−α be a positive solution of 1.1 with the initial conditions
x0 , x−1 , . . . , x−α ∈ 0, ∞. It follows from Lemma 2.1 that
a < lim inf xn m gM ≤ lim sup xn M < ∞.
2.8
Obviously, every sequence
L, gL, L, gL, . . .
2.9
is a 2-periodic not necessarily prime solution of 1.1, where L ∈ {M, m}.
By taking a subsequence, we may assume that there exists a sequence tn ≥ 2kl 1 such
that
lim xtn M,
n→∞
lim xtn −j Aj ∈ gM, M
n→∞
for j ∈ {1, 2, . . . , 2kl}.
2.10
According to 1.1, 2.10, and H3 , we obtain
f gM, M M f Al , A2k ≤ f gM, M ,
2.11
from which it follows that
Al gM,
A2k M.
2.12
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Advances in Difference Equations
In a similar fashion, we can obtain
f gM, M M A2k f A2kl , A4k ≤ f gM, M ,
f M, gM gM Al f A2l , Al2k ≥ f M, gM ,
2.13
from which it follows that
A4k A2k A2l M,
A2kl Al gM.
2.14
Inductively, we have
Aj2k M
Ajl gM
Ajl M
Ajlr2k Ajl
for j ∈ {1, 2, . . . , l},
for j ∈ {1, 3, . . . , 2k − 1},
for j ∈ {0, 2, . . . , 2k},
2.15
for j ∈ {0, 1, . . . , 2k}, r ∈ {0, 1, . . . , l}, jl r2k ≤ 2kl.
For every r ∈ {0, 1, 2, 3, . . . , 2k − 1}, there exist jr ∈ {0, 1, 2, 3, . . . , 2k − 1} and pr ∈
{0, 1, . . . , l − 1} such that jr l 2kpr r, from which, with 2.15, it follows that
A2kl−1r Ajr l M
lim xtn −2kl−1−j M
n→∞
for r ∈ {0, 2, 4, . . . , 2k − 2},
gM
2.16
for j ∈ {0, 2, . . . , 2k},
lim xtn −2kl−1−j gM
n→∞
for r ∈ {1, 3, . . . , 2k − 1},
for j ∈ {1, 3, . . . , 2k − 1}.
2.17
In view of 2.17, for any 0 < ε < M − a, there exists some tβ ≥ 4kl such that
M − ε < xtβ −2kl−1−j < M ε
gM ε < xtβ −2kl−1−j < gM − ε
if j ∈ {0, 2, . . . , 2k},
if j ∈ {1, 3, . . . , 2k − 1}.
2.18
By 1.1 and 2.18, we have
xtβ −2kl−11 f xtβ −2kl−1−l1 , xtβ −2kl1 < f M − ε, gM − ε gM − ε.
2.19
Also 1.1, 2.18, and 2.19 imply that
xtβ −2kl−12 f xtβ −2kl−1−l2 , xtβ −2kl2 > f gM − ε, M − ε M − ε.
2.20
Inductively, it follows that
xtβ −2kl−12n > M − ε
∀n ≥ 0,
xtβ −2kl−12n1 < gM − ε
∀n ≥ 0.
2.21
T. Sun and H. Xi
5
Therefore,
lim x2n M,
n→∞
lim x2n1 gM
2.22
lim x2n1 M.
2.23
n→∞
or
lim x2n gM,
n→∞
n→∞
The proof is complete.
Remark 2.2. 1 The proofs of Lemma 2.1 and Theorem 1.1 draw on ideas from the proofs of
Theorems 2.1 and 2.2 in 6.
2 Consider the nonlinear difference equation
xn1 f xn−ls1 , xn−2ks1 ,
n 0, 1, . . . ,
2.24
where s, k, l ∈ {1, 2, . . . } with 2k /
l and gcd 2k, l 1, the initial values x−α , x−α1 , . . . , x0 ∈
i
0, ∞ with α max {ls − 1, 2ks − 1}, and f satisfies H1 –H3 . Let yn1
xnsi1 for every
0 ≤ i ≤ s − 1 and n 0, 1, 2, . . . , then 2.24 reduces to the equation
i
i
i
yn1
f yn−l1
, yn−2k1
,
0 ≤ i ≤ s − 1, n 0, 1, 2, . . . .
2.25
It follows from Theorem 1.1 that for any 0 ≤ i ≤ s − 1, every positive solution of the equation
i
i
i
fyn−l1
, yn−2k1
converges to not necessarily prime a 2-periodic solution. Thus every
yn1
positive solution of 2.24 converges to not necessarily prime a 2s-periodic solution.
3. Examples
To illustrate the applicability of Theorem 1.1, we present the following examples.
Example 3.1. Consider the equation
p
xn1 m
m1
i1
i
xn−2k1
i
i0 xn−2k1
xn−l1
,
n 0, 1, . . . ,
3.1
where m, k, l ∈ {1, 2, . . . } with 2k /
l and gcd 2k, l 1 and the initial values x−α , x−α1 , . . . ,
x0 ∈ 0, ∞ with α max {l − 1, 2k − 1}, 0 < p ≤ 1. Let E 0, ∞ and
p m1
yi
fx, y m i1
i
i0 y x
x ≥ 0, y ≥ 0,
gx p
x
x > 0.
3.2
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Advances in Difference Equations
It is easy to verify that H1 –H3 hold for 3.1. It follows from Theorem 1.1 that every solution
of 3.1 converges to not necessarily prime a 2-periodic solution.
Example 3.2. Consider the equation
xn1 1 m
m1
xn−2k1
i
i1 xn−2k1
xn−l1
,
n 0, 1, . . . ,
3.3
where m, k, l ∈ {1, 2, . . . } with 2k /
l and gcd 2k, l 1 and the initial values x−α , x−α1 , . . . ,
x0 ∈ 0, ∞ with α max {l − 1, 2k − 1}. Let E 0, ∞ and
y m1
fx, y 1 m i
i1 y x
x > 0, y > 0,
gx x
x−1
x > 1.
3.4
It is easy to verify that H1 –H3 hold for 3.3. It follows from Theorem 1.1 that every solution
of 3.3 converges to not necessarily prime a 2-periodic solution.
Acknowledgments
The authors would like to thank the referees for some valuable and constructive comments
and suggestions. The project is supported by NNSF of China 10461001 and NSF of Guangxi
0640205, 0728002.
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